Stability and Unsteadiness in a 2D Laminar Shock-Induced Separation Bubble

Abstract

A computational study of the interaction between an oblique shock-wave and a laminar boundary-layer on a flat plate is carried out to investigate the unsteady character of a two-dimensional (2D) separation bubble at a free-stream Mach number of 1.5. In order to validate the code, a review of the case is presented, highlighting some significant differences compared to previous works that were not run sufficiently far in time. A steady base flow is forced with white noise and a global analysis is carried out to investigate the response of the shockinduced separation bubble. An amplitude sensitivity study is carried out for two different forcing locations, one upstream and one within the separation bubble. Independently of the forcing location all the spectra show a high frequency peak, which is demonstrated to be a Kelvin-Helmholtz instability by the linear e N -method, and an unsteady low-frequency behaviour. For low amplitude forcing applied within the bubble the low-frequency response is linear. For upstream forcing the low-frequency response is driven by the non-linearity of the downstream shedding. The same phenomenon occurs when downstream forcing is applied over a narrow band that does not include low frequencies. The occurrence of low-frequency unsteadiness in a 2D laminar problem shows the possibility to link recent works on turbulent interaction back to simpler laminar problems. In particular it is shown that the low-frequency occurs in the absence of upstream disturbances and that it can be effectively driven by broadband disturbances within the separation bubble, for example arising from weakly non-linear interaction of shear-layer modes instability.

Numerical Investigation of Shock-Induced Laminar Separation Bubble in a Mach 2 Boundary Layer

“…In the present investigation of the interaction of an oblique shock wave and a laminar boundary layer, the simulation setup follows the experiments of Hakkinen et al 1 Skin friction and wall pressure at the plate are presented in figure 3 and are compared with results from Hakkinen et al, 1 Katzer 23 and Sandham et al 13,24 As expected, the results show rapid changes near the separation and reattachment and a pressure plateau in the bubble region. The pressure ratio reaches the value of 1.4 downstream of the reflected shock in accordance with the Rankine-Hugoniot relations.…”

“…23, 24 Katzer 23 also obtained a longer bubble than in the experiments, however it is shorter than ours and those from other recent two-dimensional numerical calculation. The reasons for these differences have been investigated recently by Sansica et al 13 They also studied the effect of using different models for calculating the viscosity as a function of temperature on the bubble length. They concluded that the different methods used to calculate the viscosity does not affect the bubble length.…”

“…Our present results closely match those by Sandham and co-workers. 13 Contours of streamwise velocity, wall-normal velocity, pressure, temperature, density and Mach number are shown in figures 4 and 5. In can be observed that, as the shock wave impinges on the laminar boundary layer, it causes separation of the flow.…”

Shock-wave/boundary-layer interactions in the automatic source-code generation framework OpenSBLI

“…To validate the code implementation, comparison to the reference solution published in [1] was performed. A (609 × 255) grid stretched in the wall normal x 1 direction was used, with length L x1 and stretch factor β as…”

“…Shock-wave/boundary-layer interactions (SBLI) are a significant factor in the consideration of transonic/supersonic aircraft designs, with the induced adverse pressure gradients leading to detrimental boundary-layer thickening and separation of the flow. Recent studies of laminar SBLI include [1], [2], the first of which was simulated in the legacy Fortran code 'SBLI'. OpenSBLI [3] is a modern Python-based 'future-proof' version of the SBLI code, generating C code tailored to a user specified problem; the symbolic algebra library SymPy is used extensively throughout.…”